Several computation techniques and formulas for signature of Lefschetz fibrations on 4-manifolds exist in the literature. For Lefschetz fibrations over \({\mathbb {S}}^2\)  with fiber genus \(g=1,2\)  Matsumoto [1]}; for hyperelliptic Lefschetz fibrations over surfaces with fiber genus \(g \ge 3,\)  Endo [2]} gave signature formulas. On the other hand, Endo and Nagami [3]} showed that the signature of a Lefschetz fibration over \({\mathbb {S}}^2\)  can be calculated by using the signatures of relations contained in its monodromy. Ozbagci gave an algorithm which works for Lefschetz fibrations over \({\mathbb {D}}^2\)  or \({\mathbb {S}}^2\)  with closed fibers [4]}. Recently, Miyamura [5]} presented a formula for Lefschetz fibrations over \({\mathbb {D}}^2\)  with planar fibers. In this paper, we present an algorithm for computing the signature of Lefschetz fibrations over \({\mathbb {D}}^2\)  of any genus \(g\)  fibers which can be closed or bordered.
